Abstract:Vertex Operator Algebra (shortly VOA) is a mathematical object to study
a chiral algebra of conformal field theory from a view point of axioms.
So we consider it algebraically. In order to consider a finite type, there
are two conditions: completely reducibility and C2-finiteness
condition. As it is well observed in many finite models, the space of characters
of modules is SL2(Z)-invariant. Zhu showed that
if SW(t) is a character of
module W then SW(-1/t)
is a linear sum of characters under the above two conditions.

In this talk, we will show that C2-finiteness is sufficient
to get SL2(Z)-invariance in a sense. In this case,
we have to consider not only characters (linear sums of powers of q
= e2pit),
but also extended characters (including logarithmic forms), which are defined
by pseudo-trace functions induced from symmetric linear functions of extended
Zhu-algebra.